Source code on Github
------------------------------------------------------------------------
-- The Agda standard library
--
-- Integers, basic types and operations
------------------------------------------------------------------------

-- See README.Data.Integer for examples of how to use and reason about
-- integers.

{-# OPTIONS --cubical-compatible --safe #-}

-- Disabled to prevent warnings from deprecated names
{-# OPTIONS --warn=noUserWarning #-}

module Data.Integer.Base where

open import Data.Bool.Base using (Bool; true; false)
open import Data.Empty using ()
open import Data.Unit.Base using ()
open import Data.Nat.Base as 
  using (; z≤n; s≤s) renaming (_+_ to _ℕ+_; _*_ to _ℕ*_)
open import Data.Sign as Sign using (Sign) renaming (_*_ to _S*_)
open import Function
open import Level using (0ℓ)
open import Relation.Binary using (Rel)
open import Relation.Binary.PropositionalEquality.Core
  using (_≡_; _≢_; refl)
open import Relation.Nullary using (¬_)
open import Relation.Unary using (Pred)

infix  8 -_
infixl 7 _*_ _⊓_
infixl 6 _+_ _-_ _⊖_ _⊔_
infix  4 _≤_ _≥_ _<_ _>_ _≰_ _≱_ _≮_ _≯_
infix  4 _≤ᵇ_

------------------------------------------------------------------------
-- Types

open import Agda.Builtin.Int public
  using ()
  renaming
  ( Int    to 
  ; pos    to +_      -- "+ n"      stands for "n"
  ; negsuc to -[1+_]  -- "-[1+ n ]" stands for "- (1 + n)"
  )

-- Some additional patterns that provide symmetry around 0

pattern +0       = + 0
pattern +[1+_] n = + (ℕ.suc n)

------------------------------------------------------------------------
-- Constants

0ℤ : 
0ℤ = +0

-1ℤ : 
-1ℤ = -[1+ 0 ]

1ℤ : 
1ℤ = +[1+ 0 ]

------------------------------------------------------------------------
-- Conversion

-- Absolute value.

∣_∣ :   
 + n       = n
 -[1+ n ]  = ℕ.suc n

-- Gives the sign. For zero the sign is arbitrarily chosen to be +.

sign :   Sign
sign (+ _)    = Sign.+
sign -[1+ _ ] = Sign.-

------------------------------------------------------------------------
-- Ordering

data _≤_ :     Set where
  -≤- :  {m n}  (n≤m : n ℕ.≤ m)  -[1+ m ]  -[1+ n ]
  -≤+ :  {m n}  -[1+ m ]  + n
  +≤+ :  {m n}  (m≤n : m ℕ.≤ n)  + m  + n

data _<_ :     Set where
  -<- :  {m n}  (n<m : n ℕ.< m)  -[1+ m ] < -[1+ n ]
  -<+ :  {m n}  -[1+ m ] < + n
  +<+ :  {m n}  (m<n : m ℕ.< n)  + m < + n

_≥_ : Rel  0ℓ
x  y = y  x

_>_ : Rel  0ℓ
x > y = y < x

_≰_ : Rel  0ℓ
x  y = ¬ (x  y)

_≱_ : Rel  0ℓ
x  y = ¬ (x  y)

_≮_ : Rel  0ℓ
x  y = ¬ (x < y)

_≯_ : Rel  0ℓ
x  y = ¬ (x > y)

------------------------------------------------------------------------
-- Boolean ordering

-- A boolean version.
_≤ᵇ_ :     Bool
-[1+ m ] ≤ᵇ -[1+ n ] = n ℕ.≤ᵇ m
(+ m)    ≤ᵇ -[1+ n ] = false
-[1+ m ] ≤ᵇ (+ n)    = true
(+ m)    ≤ᵇ (+ n)    = m ℕ.≤ᵇ n

------------------------------------------------------------------------
-- Simple predicates

-- See `Data.Nat.Base` for a discussion on the design of these.

NonZero : Pred  0ℓ
NonZero i = ℕ.NonZero  i 

Positive : Pred  0ℓ
Positive +[1+ n ] = 
Positive +0       = 
Positive -[1+ n ] = 

Negative : Pred  0ℓ
Negative (+ n)    = 
Negative -[1+ n ] = 

NonPositive : Pred  0ℓ
NonPositive +[1+ n ] = 
NonPositive +0       = 
NonPositive -[1+ n ] = 

NonNegative : Pred  0ℓ
NonNegative (+ n)    = 
NonNegative -[1+ n ] = 

-- Constructors

≢-nonZero :  {i}  i  0ℤ  NonZero i
≢-nonZero { +[1+ n ]} _   = _
≢-nonZero { +0}       0≢0 = 0≢0 refl
≢-nonZero { -[1+ n ]} _   = _

>-nonZero :  {i}  i > 0ℤ  NonZero i
>-nonZero (+<+ (s≤s m<n)) = _

<-nonZero :  {i}  i < 0ℤ  NonZero i
<-nonZero -<+ = _

positive :  {i}  i > 0ℤ  Positive i
positive (+<+ (s≤s m<n)) = _

negative :  {i}  i < 0ℤ  Negative i
negative -<+ = _

nonPositive :  {i}  i  0ℤ  NonPositive i
nonPositive -≤+       = _
nonPositive (+≤+ z≤n) = _

nonNegative :  {i}  i  0ℤ  NonNegative i
nonNegative {+0}       _ = _
nonNegative {+[1+ n ]} _ = _

------------------------------------------------------------------------
-- A view of integers as sign + absolute value

infix 5 _◂_ _◃_

_◃_ : Sign    
_       ℕ.zero  = + ℕ.zero
Sign.+  n       = + n
Sign.-  ℕ.suc n = -[1+ n ]

data SignAbs :   Set where
  _◂_ : (s : Sign) (n : )  SignAbs (s  n)

signAbs :  i  SignAbs i
signAbs -[1+ n ] = Sign.-  ℕ.suc n
signAbs +0       = Sign.+  ℕ.zero
signAbs +[1+ n ] = Sign.+  ℕ.suc n

------------------------------------------------------------------------
-- Arithmetic

-- Negation.

-_ :   
- -[1+ n ] = +[1+ n ]
- +0       = +0
- +[1+ n ] = -[1+ n ]

-- Subtraction of natural numbers.
-- We define it using _<ᵇ_ and _∸_ rather than inductively so that it
-- is backed by builtin operations. This makes it much faster.
_⊖_ :     
m  n with m ℕ.<ᵇ n
... | true  = - + (n ℕ.∸ m)
... | false = + (m ℕ.∸ n)

-- Addition.

_+_ :     
-[1+ m ] + -[1+ n ] = -[1+ ℕ.suc (m ℕ+ n) ]
-[1+ m ] + +    n   = n  ℕ.suc m
+    m   + -[1+ n ] = m  ℕ.suc n
+    m   + +    n   = + (m ℕ+ n)

-- Subtraction.

_-_ :     
i - j = i + (- j)

-- Successor.

suc :   
suc i = 1ℤ + i

-- Predecessor.

pred :   
pred i = -1ℤ + i

-- Multiplication.

_*_ :     
i * j = sign i S* sign j   i  ℕ*  j 

-- Maximum.

_⊔_ :     
-[1+ m ]  -[1+ n ] = -[1+ ℕ._⊓_ m n ]
-[1+ m ]  +    n   = + n
+    m    -[1+ n ] = + m
+    m    +    n   = + (ℕ._⊔_ m n)

-- Minimum.

_⊓_ :     
-[1+ m ]  -[1+ n ] = -[1+ ℕ._⊔_ m n ]
-[1+ m ]  +    n   = -[1+ m ]
+    m    -[1+ n ] = -[1+ n ]
+    m    +    n   = + (ℕ._⊓_ m n)

------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 1.1

-- The following definition of _<_ results in the unsolved metas for the
-- first argument in certain situations.

infix  4 _<′_ _>′_ _≮′_ _≯′_

_<′_ : Rel  _
x <′ y = suc x  y
{-# WARNING_ON_USAGE _<′_
"Warning: _<′_ was deprecated in v1.1.
Please use _<_ instead."
#-}
_>′_ : Rel  _
x >′ y = y <′ x
{-# WARNING_ON_USAGE _>′_
"Warning: _>′_ was deprecated in v1.1.
Please use _>_ instead."
#-}
_≮′_ : Rel  _
x ≮′ y = ¬ (x <′ y)
{-# WARNING_ON_USAGE _≮′_
"Warning: _≮′_ was deprecated in v1.1.
Please use _≮_ instead."
#-}
_≯′_ : Rel  _
x ≯′ y = ¬ (x >′ y)
{-# WARNING_ON_USAGE _≯′_
"Warning: _≯′_ was deprecated in v1.1.
Please use _≯_ instead."
#-}